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Sat, 11 Oct 2003 10:38:04 +0100Sat, 11 Oct 2003 10:38:04 +0100On the adaptive selection of the parameter in regularization of ill-posed problemshttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1445
We study a possiblity to use the structure of the regularization error for a posteriori choice of the regularization parameter. As a result, a rather general form of a selection criterion is proposed, and its relation to the heuristical quasi-optimality principle of Tikhonov and Glasko (1964), and to an adaptation scheme proposed in a statistical context by Lepskii (1990), is discussed. The advantages of the proposed criterion are illustrated by using such examples as self-regularization of the trapezoidal rule for noisy Abel-type integral equations, Lavrentiev regularization for non-linear ill-posed problems and an inverse problem of the two-dimensional profile reconstruction.Sergei Pereverzev; Eberhard Schockpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1445Mon, 10 Nov 2003 10:38:04 +0100Morozov's Discrepancy Principle Under General Source Conditionshttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1343
In this paper we study linear ill-posed problems Ax = y in a Hilbert space setting where instead of exact data y noisy data y^delta are given satisfying |y - y^delta| <= delta with known noise level delta. Regularized approximations are obtained by a general regularization scheme where the regularization parameter is chosen from Morozov's discrepancy principle. Assuming the unknown solution belongs to some general source set M we prove that the regularized approximation provides order optimal error bounds on the set M. Our results cover the special case of finitely smoothing operators A and extends recent results for infinitely smoothing operators.M. Thamban Nair; Eberhard Schock; Ulrich Tautenhahnpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1343Thu, 29 Aug 2002 00:00:00 +0200The finite-section approximation for ill-posed integral equations on the half-linehttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1260
Integral equations on the half of line are commonly approximated by the finite-section approximation, in which the infinite upper limit is replaced by apositie number called finite-section parameter. In this paper we consider the finite-section approximation for first kind intgral equations which are typically ill-posed and call for regularization. For some classes of such equations corresponding to inverse problems from optics and astronomy we indicate the finite-section parameters that allows to apply standard regularization techniques. Two discretization schemes for the finite-section equations ar also proposed and their efficiency is studied.Sergei Pereverzev; Eberhard Schockpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1260Fri, 07 Sep 2001 00:00:00 +0200A discrepancy principle for Tikhonov regularization with approximately specified datahttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/754
Many discrepancy principles are known for choosing the parameter \(\alpha\) in the regularized operator equation \((T^*T+ \alpha I)x_\alpha^\delta = T^*y^\delta\), \(||y-y^d||\leq \delta\), in order to approximate the minimal norm least-squares solution of the operator equation \(Tx=y\). In this paper we consider a class of discrepancy principles for choosing the regularization parameter when \(T^*T\) and \(T^*y^\delta\) are approximated by \(A_n\) and \(z_n^\delta\) respectively with \(A_n\) not necessarily self - adjoint. Thisprocedure generalizes the work of Engl and Neubauer (1985),and particular cases of the results are applicable to the regularized projection method as well as to a degenerate kernel method considered by Groetsch (1990).M. Thamban Nair; Eberhard Schockpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/754Mon, 03 Apr 2000 00:00:00 +0200Toying with Jordan matriceshttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/788
It is shown that an important resolvent estimate is unstable under small perturbations.Eberhard Schockpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/788Mon, 03 Apr 2000 00:00:00 +0200Iterative Methods with Perturbations for Ill-Posed Problemshttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/789
We consider regularizing iterative procedures for ill-possed problems with random and nonrandom additive errors. The rate of square-mean convergence for iterative procedures with random errors is studied. The comparison theorem is established for the convergence of procedures with and without additive errors.Eberhard Schock; A.Yu. Veretennikovpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/789Mon, 03 Apr 2000 00:00:00 +0200On the efficient discretization of integral equations of the third kindhttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/825
We propose a new discretization scheme for solving ill-posed integral equations of the third kind. Combining this scheme with Morozov's discrepancy principle for Landweber iteration we show that for some classes of equations in such method a number of arithmetic operations of smaller order than in collocation method is required to appoximately solve an equation with the same accuracy.Sergei V. Pereverzev; Eberhard Schock; Sergei G. Solodkypreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/825Mon, 03 Apr 2000 00:00:00 +0200Brakhage's implicit iteration method and Information Complexity of equations with operators having closed rangehttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/828
An a posteriori stopping rule connected with monitoringthe norm of second residual is introduced forBrakhage's implicit nonstationary iteration method, applied to ill-posed problems involving linear operatorswith closed range. It is also shown that for someclasses of equations with such operators the algorithmconsisting in combination of Brakhage's method withsome new discretization scheme is order optimal in the sense of Information Complexity.Sergei V. Pereverzev; Eberhard Schockpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/828Mon, 03 Apr 2000 00:00:00 +0200Morozov's discrepancy principle for Tikhonov regularization of severely ill-posed problems in finite-dimensional subspaceshttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/854
Sergei Pereverzev; Eberhard Schockpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/854Tue, 08 Feb 2000 00:00:00 +0100Error estimates for band-limited spherical regularization wavelets in some inverse problems of satellite geodesyhttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/800
In this paper we discuss a special class of regularization methods for solving the satellite gravity gradiometry problem in a spherical framework based on band-limited spherical regularization wavelets. Considering such wavelets as a reesult of a combination of some regularization methods with Galerkin discretization based on the spherical harmonic system we obtain the error estimates of regularized solutions as well as the estimates for regularization parameters and parameters of band-limitation.Sergei Pereverzev; Eberhard Schockpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/800Thu, 09 Dec 1999 00:00:00 +0100